Rubber-like materials such as elastomers have been used in many parts and structures in various industries (e.g., automotive, aerospace, etc.) for years. But the mechanical properties (e.g., stress-strain or stress-stretch ratio relationship) of elastomer, other than the elastic properties, are still not clearly defined; hence, designs and analyses (especially used in computer-aided engineering) of these structures are generally based on the elastic properties of the elastomer only. In reality, elastomers exhibit non-elastic effects such as the Mullins effect, viscoelastic, and chronorheological behavior, and the magnitudes of the non-elastic properties are often large enough that they should not be neglected.
Elastomer in its virgin state exhibits a relatively stiffer response on the initial loading. When the elastomer is loaded, subsequently unloaded, then reloaded, the stress-strain relationship follows a significantly softer path. After several unloading-reloading cycles, the stress-strain relationship stabilizes, and additional unloading-reloading cycles retrace the stabilized path in the stress-strain curve. The non-elastic material behavior of elastomer described herein is referred to as the Mullins effect, in which the stress-strain relationship depends on the maximum loading previously encountered.
To date, very few analytical and experimental studies for determining non-elastic properties of elastomers have been attempted. This is because the study of mechanics for elastomers must consider both geometric and material nonlinearities. The additional effects and the lack of an adequate constitutive equation that describes these phenomena make the analytical and experimental studies very difficult.
The determination of the material properties is generally conducted using uni-axial test, as shown in FIG. 1A, in which an elastomer specimen 102 is pulled by a uni-axial tension 104 (i.e., one-dimensional test) at either end. However, the material properties obtained from the one-dimensional test may not represent true behaviors of elastomer contained in a structure, which is generally not in a one-dimensional space. To solve this deficiency, one of the prior art attempts is to stretch a sheet of an elastomer specimen (membrane) 112 in two planar directions as shown in FIG. 1B. The specimen 112 is pulled by equal tensions 114 in both of the specimen's planar axes. However, this prior art approach requires a huge piece of specimen (thereby huge laboratory) to avoid the edge effects from point loads (i.e., tensions 114) applied around the perimeter. This is not a practical solution because the test would have to be conducted in a very large test facility or laboratory.
Further, the current or prior art numerical equations (i.e., constitutive equations) are not adequate to represent true behaviors of elastomers. One of the prior art constitutive equations is the Ogden equation, which suggests or assumes the loading and subsequent reloading paths are the same. The Ogden equation does not represent the true behaviors of elastomers, which become softer in a subsequent reloading path than in the original loading path.
The elastomer behaviors as shown in FIG. 2 are calculated using the Ogden equation. In FIG. 2, the vertical axis represents stress in the elastomer. The horizontal axis represents the stretch ratio λ of the elastomer. The stretch ratio is defined as the stretched length divided by the original length of the elastomer. Therefore, the relationship between the stretch ratio λ and the strain ε is that ε=λ−1. The elastomer is first loaded following the path 202 starting at un-stretched position 201 (i.e., stretch ratio equal to one) until the stretch ratio reaches four (4) at 203. Then the elastomer is unloaded following a first unload path 204 back to the origin 201. Next the elastomer is reloaded following the reloading path 212 to a stretch ratio about 5.5 at 213. The first portion of the reloading path 212 is assumed to be the same as the original loading path 202 in the Ogden equation. It is emphasized that this is an incorrect assumption thus leading to inaccurate numerical analysis of elastomers. Finally the elastomer is unloaded again at 213. Based on the Ogden equation, the elastomer follows the second unloading path 214 back to the origin 201.
Given the foregoing drawbacks, problems and limitations of the prior art, it would be desirable to have improved methods and systems to determine material properties of elastomers such that the more correct behaviors of elastomers such as the Mullins effect can be numerically calculated and analyzed.